Galois Theory Tom Leinster, University of Edinburgh Version of 13 May 2021 Note to the reader2 1 Overview of Galois theory4 1.1 The view of C from Q ........................ 4 1.2 Every polynomial has a symmetry group. ............. 9 1.3 . which determines whether it can be solved........... 11 2 Rings and fields 14 2.1 Rings ................................ 14 2.2 Fields ................................ 19 3 Polynomials 24 3.1 The ring of polynomials....................... 24 3.2 Factorizing polynomials ...................... 28 3.3 Irreducible polynomials....................... 33 4 Field extensions 38 4.1 Definition and examples ...................... 38 4.2 Algebraic and transcendental elements............... 43 4.3 Simple extensions.......................... 47 5 Degree 51 5.1 Degrees of extensions and polynomials............... 51 5.2 The tower law............................ 56 5.3 Algebraic extensions ........................ 58 5.4 Ruler and compass constructions.................. 60 6 Splitting fields 67 6.1 Extending homomorphisms..................... 68 6.2 Existence and uniqueness of splitting fields ............ 70 6.3 The Galois group.......................... 76 7 Preparation for the fundamental theorem 82 7.1 Normality.............................. 83 7.2 Separability............................. 91 7.3 Fixed fields ............................. 96 8 The fundamental theorem of Galois theory 100 8.1 Introducing the Galois correspondence . 100 8.2 The theorem.............................104 8.3 A specific example .........................109 9 Solvability by radicals 115 9.1 Radicals...............................116 9.2 Solvable polynomials have solvable groups . 119 9.3 An unsolvable polynomial .....................126 10 Finite fields 130 10.1 ?th roots in characteristic ? . 131 10.2 Classification of finite fields ....................133 10.3 Multiplicative structure.......................135 10.4 Galois groups for finite fields....................136 1 Note to the reader These are the 2020–21 course notes for Galois Theory (MATH10080). Structure Each chapter corresponds to one week of the semester. You should read Chapter 1 in Week 1, Chapter 2 in Week 2, and so on. I’m writing the notes as we go along, so the chapters will appear one by one: keep your eye on Learn. Exercises looking like this are sprinkled through the notes. The idea is that you try them immediately, before you continue reading. Most of them are meant to be quick and easy, much easier than as- signment or workshop questions. If you can do them, you can take it as a sign that you’re following. For those that defeat you, talk with your group or ask on Piazza; or if you’re really stuck, ask me. I promise you that if you make a habit of trying every exercise, you’ll enjoy the course more and understand it better than if you don’t bother. Digressions like this are optional and not examinable, but might interest you. They’re usually on points that I find interesting, and often describe connections between Galois theory and other parts of mathematics. Here you’ll see titles of relevant References to theorem numbers, page numbers, etc., are clickable links. videos What to prioritize You know by now that the most important things in almost any course are the definitions and the results called Theorem. But I also want to emphasize the proofs. This course presents a wonderful body of theory, and the idea is that you learn it all, including the proofs that are its beating heart. A closed-book exam would test that by asking you to reproduce some proofs. Your exam will be open book, so it can’t ask you to reproduce proofs, but it will test something arguably harder: that you understand them. So the proofs will need your attention and energy. 2 Prerequisites You are required to have taken these two courses: • Honours Algebra: We’ll need some linear algebra, corresponding to Chap- ter 1 of that course. For example, you should be able to convince yourself that an endomorphism of a finite-dimensional vector space is injective if and only if it is surjective. We’ll also need everything from Honours Algebra about rings and polyno- mials (Chapter 3 there), including ideals, quotient rings (factor rings), the universal property of quotient rings, and the first isomorphism theorem for rings. • Group Theory: From that course, we’ll need fundamentals such as normal subgroups, quotient groups, the universal property of quotient groups, and the first isomorphism theorem for groups. You should know lots about the symmetric groups (=, alternating groups =, cyclic groups = and dihedral groups =, and I hope you can list all of the groups of order < 8 without having to think too hard. Chapter 8 of Group Theory, on solvable groups, will be crucial for us! If you skipped it, you’ll need to go back and fix that. For example, you’ll need to understand the statement that (4 is solvable but 5 is not. We won’t need anything on free groups, the Sylow theorems, or the Jordan– Hölder theorem. If you’re a visiting student and didn’t take those courses, please get in touch so we can decide whether your background is suitable. Mistakes I’ll be grateful to hear of any mistakes ([email protected]), even if it’s something very small and even if you’re not sure. 3 Chapter 1 Overview of Galois theory This chapter stands apart from all the others, Modern treatments of Galois theory take advantage of several well-developed branches of algebra: the theories of groups, rings, fields, and vector spaces. This is as it should be! However, assembling all the algebraic apparatus will take us some time, during which it’s easy to lose sight of what it’s all for. Introduction to Galois theory came from two basic insights: Week 1 • every polynomial has a symmetry group; • this group determines whether the polynomial can be solved by radicals (in a sense I’ll define). In this chapter, I’ll explain these two ideas in as short and low-tech a way as I can manage. In Chapter2 we’ll start again, beginning the modern approach that will take up the rest of the course. But I hope that all through that long build-up, you’ll keep in mind the fundamental ideas that you learn in this chapter. 1.1 The view of C from Q Imagine you lived several centuries ago, before the discovery of complex numbers. Your whole mathematical world is the real numbers, and there is no square root of −1. This situation frustrates you, and you decide to do something about it. So, you invent a new symbol 8 (for ‘imaginary’) and decree that 82 = −1. You still want to be able to do all the usual arithmetic operations (¸, ×, etc.), and you want to keep all the rules that govern them (associativity, commutativity, etc.). So you’re also forced to introduce new numbers such as 2 ¸ 3 × 8, and you end up with what today we call the complex numbers. So far, so good. But then you notice something strange. When you invented the complex numbers, you only intended to introduce one square root of −1. But 4 accidentally, you introduced a second one at the same time: −8. (You wait centuries for a square root of −1, then two arrive at once.) Maybe that’s not so strange in itself; after all, positive reals have two square roots too. But then you realize something genuinely weird: There’s nothing you can do to distinguish 8 from −8. Try as you might, you can’t find any reasonable statement that’s true for 8 but not −8. For example, you notice that 8 is a solution of 17 I3 − I2 − I − = , 3 16 3 I but then you realize that −8 is too. Of course, there are unreasonable statements that are true for 8 but not −8, such as ‘I = 8’. We should restrict to statements that only refer to the known world of real numbers. More precisely, let’s consider statements of the form ? ¹Iº ? ¹Iº 1 = 3 , (1.1) ?2¹Iº ?4¹Iº where ?1,?2,?3,?4 are polynomials with real coefficients. Any such equation can be rearranged to give ?¹Iº = 0, where again ? is a polynomial with real coefficients, so we might as well just consider statements of that form. The point is that if ?¹8º = 0 then ?¹−8º = 0. Let’s make this formal. Definition 1.1.1 Two complex numbers I and I0 are indistinguishable when seen from R, or conjugate over R, if for all polynomials ? with coefficients in R, ?¹Iº = 0 () ?¹I0º = 0. Warning 1.1.2 The standard term is ‘conjugate over R’. ‘Indistin- guishable’ is a term I invented for this chapter only, to express the idea of not being able to tell the two numbers apart. For example, 8 and −8 are indistinguishable when seen from R. This follows from a more general result: Lemma 1.1.3 Let I, I0 2 C. Then I and I0 are indistinguishable when seen from R if and only if I0 = I or I0 = I. 5 Proof ‘Only if’: suppose that I and I0 are indistinguishable when seen from R. Write I = G ¸ 8H with G, H 2 R. Then ¹I − Gº2 = −H2. Since G and H are real, indistinguishability implies that ¹I0 − Gº2 = −H2, so I0 − G = ±8H, so I0 = G ± 8H. ‘If’: obviously I is indistinguishable from itself, so it’s enough to prove that I is indinguishable from I. I’ll give two proofs. Each one teaches us a lesson that will be valuable later. First proof: recall that complex conjugation satisfies F1 ¸ F2 = F1 ¸ F2, F1 · F2 = F1 · F2 for all F1,F2 2 C, and 0 = 0 for all 0 2 R. It follows by induction that for any polynomial ? over R, ?¹Fº = ?¹Fº for all F 2 C.